Precoloring extension for K4-minor-free graphs

Let G = (V, E) be a graph where every vertex v ∈ V is assigned a list of available colors L(v). We say that G is list colorable for a given list assignment if we can color every vertex using its list such that adjacent vertices get different colors. If L(v) = {1, . . . , k} for all v ∈ V then a corresponding list coloring is nothing other than an ordinary k-coloring of G. Assume that W ⊆ V is a subset of V such that G[W ] is bipartite and each component of G[W ] is precolored with two colors. The minimum distance between the components of G[W ] is denoted by d(W ). We will show that if G is K4-minor-free and d(W ) ≥ 7, then such a precoloring of W can be extended to a 4-coloring of all of V . This result clarifies a question posed in [10]. Moreover, we will show that such a precoloring is extendable to a list coloring of G for outerplanar graphs, provided that |L(v)| = 4 for all v ∈ V \W and d(W ) ≥ 7. In both cases the bound for d(W ) is best possible.